But this is not about clustering. It's about figuring out to what extent a certain subclass of features, namely the 'shapelets', are statistically significantly associated with a pre-defined binary outcome.
The paper you mentioned is interesting, though, because it shows an issue that many algorithms are privy to: if the number of samples/features gets too large, at some point, you are only comparing _means_.
(We are working on a paper to show the issues of this when it comes to time series classification.)
Seems to be standard terminology for time series classification to me, to be honest. I think the approach would also work if there are duplicates in the data. Although the estimate would be overly optimistic, right?
With their notation they have not specified that the T's are unique. So, a first fix up would be just to state that the T's were distinct. And it would help to be explicit that i from 0, 1, 2, ... corresponded to increasing time. Moreover, is the data equally spaced in time? Likely, yes, and in that case, clearly say so.